Novel implementation of a phase-only spatial light modulator for laser beam shaping

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by

Liesl Burger

Dissertation presented for the degree of Doctor of Science in Physics in the Faculty of Physics at Stellenbosch University

Department of Laser Physics, University of Stellenbosch,

Private Bag X1, Matieland 7602, South Africa.

Promoter: Prof. Andrew Forbes, Dr. Igor Litvin, Prof. Erich G Rohwer

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Declaration

By submitting this dissertation electronically, I declare that the entirety of the work con-tained therein is my own, original work, that I am the sole author thereof (save to the extent explicitly otherwise stated), that reproduction and publication thereof by Stellenbosch Uni-versity will not infringe any third party rights and that I have not previously in its entirety or in part submitted it for obtaining any qualification.

March 2016

Date: . . . .

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Abstract

Novel implementation of a phase-only spatial light modulator for laser beam shaping

L. Burger

Department of Laser Physics, University of Stellenbosch,

Private Bag X1, Matieland 7602, South Africa.

Dissertation: PhDPhys December 2015

The phase-only spatial light modulator (SLM) has revolutionized the field of laser beam shaping. In this thesis we describe in detail the considerations necessary to build a “digital laser” which incorporates an SLM into a laser cavity as an intracavity element to dynam-ically generate a wide variety of custom beams. We then present a theoretical analysis of the healing of petal-like (or Laguerre-Gaussian) beams using rotational considerations, and using digital laser technology to demonstrate this healing experimentally. We extend our investigation into self-healing beams with the theoretical derivation of a new type of Bessel-like beams, which retains a concentric ring structure on propagation, and which self-heal axially. These beams are generated using an SLM, and self-healing is demonstrated exper-imentally.

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Uittreksel

Nuwe implementering van ’n fase-alleen ruimtelike ligmodulator vir laserbundelvorming

(“Novel implementation of a phase-only spatial light modulator for laser beam shaping”)

L. Burger

Departement van laserfisika, Universiteit van Stellenbosch, Privaatsak X1, Matieland 7602, Suid Afrika.

Proefskrif: (PhDPhys) Desember 2015

Die fase-alleen ruimtelike ligmodulator (RLM) het die veld van laserbundelvorming re-volusionêr verander. In hierdie tesis beskryf ons sorgvuldig wat die nodige oorwegings is om ’n “digitale laser” te bou, wat ’n RLM in die laserholte as in ’n intra-holte element bevat, sodat dit ’n wye verskeidenheid van laserbundel vorms dinamies kan genereer. Voorts ver-skaf ons ’n teoretiese analise van die korreksie van blom-patroon (of Laguerre-Gauss) bun-dels, met die hulp van rotasie beginsels, en ons gebruik die digitale laser tegnologie om hier-die korreksie eksperimenteel te demonstreer. Ons brei ons ondersoek na self-korregerende bundels uit met die teoretiese afleiding van ’n nuwe soort Bessel-tipe bundel wat ’n konsen-triese ringstruktuur behou tydens voortplanting en wat aksiaal self-korregerend is. Hierdie bundels is met behulp van ’n RLM gegenereer en self-korreksie is eksperimenteel gede-monstreer.

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Acknowledgements

I am thankful to my supervisors Dr Andrew Forbes and Dr Igor Litvin for their patience, encouragement and excellent academic guidance. I am also grateful to the National Laser Centre which facilitated and sponsored my post-graduate studies. Thanks to my colleagues Dr Angela Dudley, Dr Darryl Naidoo and Ms Thandeka Mhlanga, who assisted with dis-cussions and in the lab. Thanks in particular to Dr Darryl Naidoo also for the considered and very helpful proof-reading of my manuscript. Thanks also to Prof. Ian Underwood of the University of Edinburgh for his interest in my work and assistance with SLM design and behaviour, and to Prof. Helen Gleeson OBE of the University of Leeds for assistance with liquid crystal material properties. Thanks especially to my family, Herman, Paul and Disa, for their love and encouragement, even if they do call me “Dr Monkey”. Thanks also to friends and family who offered support and sustenance, a list which includes Margaret Haines, Adrian & Bonnie Haines, Joyce Burger & Johan Pretorius, and Sandra & Glen McGavigan.

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Dedications

To my family: Herman, Paul & Disa

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List of Figures

1.1 Fabry-Perot resonator with spherical mirrors. . . 4 1.2 Plot of the stability function as a function of g1 (x-axis) and g2 (y-axis), with

stability in the coloured areas. . . 5 1.3 A Gaussian beam profile showing the beam radius w. . . 5 1.4 Propagation of a Gaussian laser beam. . . 6 1.5 An array of plots of the TEM HG intensity distributions where the mode indices

correspond to the horizontal, n and vertical, m index, respectively. . . 9 1.6 The cross-section of a HG00(blue), HG01(purple) and HG02(yellow) mode, all

with the same w in 1.1.11. . . 9 1.7 An array of plots of the LG intensity distributions where the mode indices

cor-respond to the radial, p and azimuthal, ` index, respectively. . . 11 1.8 Transmission values for several TEMnmmodes with w = 0.023 cm as a function

of the aperture size a. The vertical lines represent a = w, 1.5w, 2w, 2.5w, and 3w respectively. . . 12 1.9 Schematic of liquid crystal molecule, showing the origin of birefringence β . . . 14 1.10 Two examples of LC alignment schemes. (a) twisted nematic, (b) parallel aligned. 15 1.11 Typical OASLM layout, showing that the phase pattern is written to the detector

of the OASLM with “write light”, and is imparted onto the coherent “read light” by the modulator [1]. . . 16

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1.12 Structure of an EASLM, showing a liquid crystal layer (c) sandwiched between two alignment layers (d) on an aluminium pixel array (b) which is mounted on a CMOS chip (a). A voltage between the CMOS chip and in indium tin oxide layer (e) controls the birefringence of the liquid crystals in each pixel, which changes the phase of the incident light (g) to obtain modulated, reflected light (h) [1]. . . 18 1.13 Common experimental setup of an SLM to generate custom beams. Laser light

is polarized and then expanded onto the SLM. One diffraction order is selected using an iris at the focal plane of a lens and recorded on a camera (CAM). . . . 19

2.1 Theoretically calculated curves show the transmission through a TN-LC layer for twist angle 90 °, for incident polarizer angles 0°; 5°; 10° (solid line), 15° and 20°. The solid line shows the configuration with the most constant transmission [109]. . . 26 2.2 Measured reflectivities of TN-LC SLM and PA-LC SLM as a function of phase,

for vertical polarization. . . 29 2.3 Plots of normalized relative intensity as a function of phase after n = 0, 5, 10 and

20 round-trip reflections of a beam with stable mode off an SLM in a resonator, using measured intensity modulation data for (a) a TN-LC SLM and (b) a PA-LC SLM. . . 30 2.4 (a) A simple, conventional resonator comprising two reflective mirrors, (b)

Mir-ror M2replaced with a flat mirror and lens L2, (c) Flat mirror M2replaced with

blank-phase SLM, (d) Phase on SLM to simulate lens L2. . . 32

2.5 Graph of the 75 W Jenoptik (JOLD 75 CPXF 2P W) multimode fibre-coupled laser diode output power as a function of current. . . 33 2.6 Diagram of the SLM diode-pumped Nd:YAG laser. The Nd:YAG crystal was

end-pumped by an 808 nm diode laser pump (DLP). The resonator has a spatial light modulator (SLM) as back-reflector, and a 95% flat output coupler (OC). The resonator also contains a 60% flip-up mirror (FUM) in front of the SLM, a Brewster window (BW) and a 4x beam-expanding telescope (BET). . . 33

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2.7 Beam patterns produced by the laser containing the intracavity TN-LC SLM are shown in the top row, with the corresponding bitmaps below each. (a) is a Gaussian beam; (b) is a Hermite-Gauss beam (n = 1, m = 0); (c) is an 8-petal patterned beam, and (d) is a donut beam. . . 35 2.8 Examples of laser modes produced by the laser. In each case (a - h) the

field pattern is shown, with the far-field pattern inset. Notice that the near-field beam pattern matches the far-near-field pattern. The modes can be identified as: (a) Gaussian; (b) Hermite-Gauss (n = 0, m = 1); (c) Hermite-Gauss (n = 1, m = 0); (d) donut mode; (e) Hermite-Gauss (n = 0, m = 2); (f) Laguerre-Gauss (p = 0, l = ±2); (g) Laguerre-Laguerre-Gauss (p = 0, l = ±3); (h) Laguerre-Laguerre-Gauss (p = 0, l = ±4). . . 35 2.9 The set of phase screens for modal decomposition with topological charge -1 to

-5 along the bottom row, and +1 to +5 along the top row. . . 36 2.10 Modal decomposition of the 6-petal output pattern confirmed that it comprises

a superposition of Laguerre-Gauss (p = 0, l = +3) and Laguerre-Gauss (p = 0, l = −3) modes. . . 36 2.11 Changing the curvature C (where C = 1/R) on the digital holograms on the

PA-LC SLM has the effect of changing the beam waist size on the output coupler. 37 2.12 Beam patterns produced by the second prototype containing the intracavity

PA-LC SLM, identified as (a) a circular flat-top beam; (b) an Airy beam; (c) a Laguerre-Gauss beam (p = 1, l = ±2); and (d) a Laguerre-Gauss beam (p = 1, l = 0). The corresponding digital holograms are shown below each beam. Detail of the insert of digital hologram (c) is shown below to illustrate the use of a complex amplitude modulation technique (here using a checker-board pattern) to modulate amplitude in addition to phase. . . 38 3.1 Sketch of a Porro or roof prism, showing a correctly aligned prism on the left,

and a misaligned prism with vertical offset δ corresponding to an angular offset β on the right. . . 44 3.2 Schematic diagram of a Porro prism resonator, showing the following

opti-cal elements: (a, h) Porro prisms, (b, g) lenses, (c) polarizing beam cube, (d) quarter-wave plate, (e) Q-switch, (f) Nd:YAG rod. . . 44

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3.3 Output of the numerical model of Porro prism laser showing examples of beams produced with (a) Porro angle α = π

2, (b) α = π

3, (c) α = π

4, (d) α = 0.7625. . 46

3.4 Output of the numerical model of Porro prism laser showing examples of beams produced with (a) Porro angle α = 0.174, NF = 0.371, (b) α = 0.523, NF =

0.428, (c) α = 0.523, NF = 0.269, (d) α = 0.523, NF = 0.306. . . 46

3.5 (a) Schematic representation of the rotation of the shadow region in an ob-structed LG beam with different sign of the angular momentum. (b) A schematic for the derivation of the self-reconstruction distance zr, and (c) the dependence

of the reconstruction distance on initial position (zI) and angular size (θI) of

the obstacle. (d) The dependence of the maximum angular size obstruction θ (zI)maxon the initial position of the obstacle zIfor the different Rayleigh range

of the beam. . . 49 3.6 The modal decomposition of 8 petal beam. . . 50 3.7 (a) The simulation of the free space propagation of obstructed LG04and LG0−4

beams. (b) The simulation and corresponding experimental verification of the reconstruction of the superposition beam (LG04and LG0−4). . . 51

4.1 A Bessel beam (B) formed by passing a Gaussian beam (G) through an axicon (A). . . 54 4.2 Examples of Bessel beams generated by plotting Eq. 4.1.1 with n = 0, 1 and 2.

Notice that the beam of order 0 has a central peak but that higher orders have a central null. . . 55 4.3 A sequence of BLBs showing the effects of the parameter a can be seen. (a)

a= 0.0001, (b) a = 0.001, (c) a = 0.05, (d) a = 0.01 (n = 1, m = 2). . . 58 4.4 A longitudinal cross-section of the intensity distribution of a BLB

illustrat-ing the derivation of the self-reconstruction distance for BLBs. An obstruc-tion with radius r0is located at z on the optical axis OC at position AB.

Self-reconstruction occurs in the zone with length zr. . . 59

4.5 Phase screen generated using the approach outlined in this section, and used for reconstruction experiments following. n = 1, m = 2, a = 0.05 . . . 61

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4.6 Dependence of self-reconstruction distance zr(for certain values of n and m of

the transformation system) on distance to obstruction z (see Fig. 4.4) for the following parameters of initial field and system: w = 2 mm, r0= w/3, rI= 3w,

a= 3 × 10−3; (black) n = 1; m = 2; (red) n = 2, m = 3; (blue) n = 1, m = 3. . . 62 4.7 The experimental setup consists of an expanded HeNe beam reflected off the

phase screen displayed on an SLM, creating a BLB with n = 1, m = 2. An obstruction OBST (either a bead or thin wire) was positioned at a distance of z from the phase screen. A 4-f imaging system transfers the object plane OBJ to the image plane IMG on the camera sensor CAM at several axial positions zI. . 62

4.8 (a) Unobstructed BLB (n = 1, m = 2, w = 1.7 mm, a = 0.05) at the obstruction plane (zI = 0), (b) BLB at the same plane but obstructed by a centred 400µm

bead, (c) unobstructed BLB at zI= 110 mm, and (d) obstructed BLB at zI= 110

mm. . . 63 4.9 (a) unobstructed BLB, (b) obstructed BLB, (c) partial reconstruction showing

diffraction. . . 64 4.10 Beams shown at increasing distances zIfrom the obstruction of four

reconstruc-tion experiments: (a) off-centre wire, (b) off-centre bead, (c) centred wire, and (d) centred bead. In each case the calculated shadow pattern is shown (inset) for n = 1, m = 2, w = 1.7 mm, and a = 0.05 and rI= 2 mm. . . 65

4.11 The self-reconstruction of the same beam with a wire obstruction placed off-centre at (a) 248 mm, and (b) 748 mm. (c) shows that the beam is partially reconstructed 100 mm after (a), contrasted with (d) which shows very little reconstruction 100 mm after (b). (e) shows complete reconstruction of the ob-scured area at 200 mm after (a), but (f) shows that complete reconstruction is only evident at 400 mm after (b). . . 66

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List of Tables

1.1 Comparison of specifications of some commercially-available OASLMs. . . 17 1.2 Comparison of specifications of some commercially-available EASLMs. . . 18 2.1 Comparison of typical specifications of the older type of SLM using TN-LCs

and the newer type using PA-LCs. . . 27 4.1 The cone angle, θ (z), of BLBs for three example cases: n = 1, m = 2 (an

axicon-lens doublet), n = 2, m = 3 (an aberrated lens) and n = 1, m = 3 (an aberrated axicon). . . 60 4.2 The self-reconstruction distance zr for example values of n and m of the

trans-formation system. . . 61

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List of Publications

Peer-reviewed Journal Papers:

1. Litvin, I. A., Burger, L. and Forbes, A.: Angular self-reconstruction of petal-like beams. Optics letters, vol. 38, p. 3363, 2013 (see Chapter 3).

2. Ngcobo, S., Litvin, I., Burger, L. and Forbes, A.: A digital laser for on-demand laser modes. Nature communications, vol. 4, p. 2289, 2013 (see Chapter 2).

3. Ngcobo, S., Litvin, I., Burger, L. and Forbes, A.: Demonstrating a Rewritable Digital Laser. Optics and Photonics News, vol. 24, p. 28, 2013 (see Chapter 2).

4. Burger, L., Litvin, I., Ngcobo, S. and Forbes, A.: Implementation of a spatial light modulator for intracavity beam shaping. Journal of Optics, vol. 17, p. 015604, 2015 (see Chapter 2).

5. Litvin, I., Burger, L. and Forbes, A.: Self-healing of Bessel-like beams with longi-tudinally dependent cone angles. Journal of Optics, vol. 17, p. 105614, 2015 (see Chapter 4).

Patents:

1. Ngcobo, S., Litvin, I. A., Burger, L. and Forbes, A.: Method of operating a laser and laser apparatus using intra-cavity digital holograms. Patent Application US20150009547, 2014.

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International Conference Papers:

1. Burger, L., Litvin, I. and Forbes, A.: Intra-cavity beam control: a comparison of spatial light modulators and adaptive mirrors. AOIM Droplets in Murcia, 2013. 2. Burger, L., Litvin, I., Ngcobo, S. and Forbes, A.: How to make a digital laser. Third

Conference on Sensors, MEMS and Electro-Optic Systems, p. 925706, 2014.

3. Forbes, A., Ngcobo, S., Burger, L. and Litvin, I. A.: The digital laser: on-demand laser modes with the click of a button. Proceedings of SPIE, p. 89601K, 2014. 4. Ngcobo, S., Litvin, I., Burger, L. and Forbes, A.: Digital control of laser modes with

an intra-cavity spatial light modulator. Proceedings of SPIE, p. 89601X, 2014. 5. Burger, L., Litvin, I., Ngcobo, S. and Forbes, A.: Intracavity beam shaping using an

SLM. Proceedings of SPIE, p. 95810A, 2015.

National Conference Papers:

1. Burger, L., Litvin, I. and Forbes, A.: Digital mode selection using an intracavity SLM. 57th Annual Conference of the South African Institute of Physics, 2012.

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Nomenclature

List of symbols

e = 2.718 π = 3.142 c= 3 × 108ms−1 λ wavelength. . . [ m ] L resonator length . . . [ m ] R1, R2 mirror radius . . . [ m ]

g1, g2 resonator stability parameters . . . [ ]

I intensity . . . [ Wm−2]

w 1/e2_{beam radius} _{. . . .} _{[ m ]}

w₀ Gaussian beam waist . . . [ m ]

w₁, w2 beam radius on mirror . . . [ m ]

r radial coordinate . . . [ m ] z axial coordinate . . . [ m ] zR Rayleigh range . . . [ m ] C curvature . . . [ m−1] θ divergence . . . [ radians ] NF Fresnel number . . . [ ] a aperture radius . . . [ m ] M2 quality factor . . . [ ] xxiii

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H Hermite polynomial . . . [ ]

n, m order in the x, y directions. . . [ ]

L|`|p Laguerre polynomial . . . [ ]

p radial mode index. . . [ ]

` azimuthal mode index . . . [ ]

α prism rotation angle . . . [ radians ]

N number of petals . . . [ ]

ne, no extraordinary, ordinary refractive index . . . [ ]

β birefringence . . . [ ]

R reflectivity . . . [ ]

f focal length . . . [ m ]

Acronyms/Abbreviations

DOE diffractive optical element SLM spatial light modulator TEM transverse electromagnetic

HG Hermite-Gaussian

LG Laguerre-Gaussian

BG Bessel-Gaussian

LCD liquid crystal display

CMOS complementary metal oxide semiconductor OASLM optically activated SLM

EASLM electrically activated SLM

ITO indium tin oxide

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TN-LC twisted nematic liquid crystal PA-LC parallel aligned liquid crystal

OC output coupler

BW Brewster window

DLP diode laser pump

BET beam-expanding telescope

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Chapter 1

Introduction

Laser beam shaping is a dynamic and vibrant field of study which deals with the selection and manipulation of laser modes and with the modification of existing beams to create new patterns with particular phase and intensity properties. The earliest beam-shaping methods were aimed at simply achieving a Gaussian beam profile, which is the preferred output beam for many industrial materials processing applications like cutting and welding [2] because it has a low divergence and can be focussed to a very small spot, and achieved by resonator designs which limit the transverse extent of the beam while extracting maximum energy [3]. The study of the modes which form in laser resonators has led to amplitude and phase masking techniques and gain shaping techniques which allow the selection of particular chosen transverse modes with their characteristic phase and intensity distributions [4; 5; 6]. Phase modulation masks were used to create custom output intensity profiles inside the resonator [7], and using a holographic technique to shape Gaussian beams to form custom intensity profiles outside the resonator [8]. (See Section 1.3 for more on beam-shaping techniques.)

Diffractive optical elements (DOEs) and phase-only spatial light modulators (SLMs) are two common methods of phase modulation for laser beam shaping. A DOE has a phase pattern etched onto a glass substrate, and is tailored for a specific laser configuration and output pattern. DOEs have the disadvantage that a master DOE is expensive to manufacture, although it can be used to make many inexpensive copies, such as the type commonly distributed with laser pointers. A phase-only SLM allows digitally generated phase patterns to be displayed on a pixelated liquid crystal display panel controlled by a computer, in

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order to dynamically generate custom phase patterns. The invention of this device has revolutionised the field of holographic laser beam shaping, with new beams with specific properties being continuously discovered [9; 10; 11; 12; 13; 14]. The work covered in this thesis highlights the role of the phase-only SLM in laser beam shaping, first as an intracavity phase modulating device to dynamically generate a wide variety of custom beams (see Chapter 2), and in the generation of new beams with self-healing properties (see Chapters 3 and 4).

The development of the phase-only SLM device started when a new material (cholesteryl benzoate) with a mesophase between the liquid and solid state (at a certain temperature range) was discovered by an Austrian botanist, Friedrich Reinitzer in 1888. The follow-ing year Otto Lehmann, a German Professor of Physics, studied the material and found that it had a double refraction effect characteristic of crystals, and called it a "liquid crys-tal". The material remained a scientific curiosity with very little research into the material properties until 1962 when Richard Williams discovered interesting electro-optical char-acteristics. This subsequently led to the development by George H. Heilmeier of the first LCD screens, which were first used in 1972 [15]. By 1975 the dynamic switching of ne-matic LCs (see Chapter 1.2) in an electric field was well understood [16]. LC SLMs were developed in the later 1980s, emerging from the development of the LC television (TV) [17] which was developed in the late 1980s and early 1990s [18; 19; 20]. They were used first for amplitude modulation [21], and then for phase modulation [22], and for cross-coupled amplitude and phase modulation [23]. LC SLMs were low-cost, proof-of-concept devices, with typically 100 × 100 pixels, a pixel size of 100µm and a frame rate of 20kHz [24]. Two types of SLM devices had emerged: the optically activated (OA) SLM [25; 26], and the electrically activated (EA) SLM. The development of both occurred concurrently, but the OA SLM suffered from the drawback of slow switching speeds, and the development of EA SLMs was favoured. This preference accelerated with the integration of the SLM with a silicon chip [27], know as LCOS (liquid crystal on silicon) technology. Electrical activation was simpler and more efficient than optical activation, and the resolution quickly increased to 256 × 256 pixels [28], and to 1024 × 768 pixels by 2004 [29]. At around this time, with the implementation of new liquid crystal material [30], phase-mostly SLMs became truly phase-only. This is discussed in more detail in Chapter 2.2. Modern phase-only LCOS

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SLM devices can have up to 1920 × 1200 pixels with pixel size 4µm, and run at 60 Hz [31]. The first application of LC SLMs was 3D holography [32; 33; 34; 35]. Soon after, computer-generated holograms were created and used for rudimentary beam shaping of Fresnel lenses [18; 36; 37] and for static [38] and dynamic [39] mirror tilt. It was shown that a phase-only SLM can produce a large set of Zernike polynomials [40], and this was demonstrated in real-time[41], making them suitable for adaptive optical wave-front correc-tion [30] in applicacorrec-tions such as medical imaging [42] and terrestrial telescopic systems to correct for atmospheric aberration [43; 44]. The biggest impact of these devices, however, has been in the field of laser beam shaping. Phase-only SLMs made possible the creation of a wide range of novel laser beams, most of which cannot be created in any other way [45; 46; 47; 48; 10; 11]. Novel beams created by SLMs form the basis of the field of laser tweezing [49; 50; 51], have been used in the study of atmospheric aberrations [52], and have been used for commercial applications such as laser marking [53] and micro-machining [54].

This thesis presents new examples of how phase-only SLMs are advancing the field of laser beam shaping. The first example is of a new invention, the “digital laser” [55; 56; 57; 58; 59; 60], which arose out of a need to fine-tune the design of intracavity diffractive optical elements. This work uncovered subtle properties of SLMs that become dominant when used in an intracavity configuration, and led to a device which is able to dynamically generate a wide range of laser modes and beam shapes. The second and third examples use an SLM to generate beams with different regeneration properties, the mechanisms of which are presented along with experimental results [61; 62].

1.1 Basic laser theory

1.1.1 Laser resonators

An optical resonator is a series of optical components that allows laser light to circulate. The simplest and most common configuration is the Fabry-Perot resonator, which consists of two spherical mirrors: a fully reflective back reflector and a partially reflective output coupler.

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R

1

_R

2

L

2ù

₀

z

2

z

1

2ù

₁

_2ù

2

Figure 1.1: Fabry-Perot resonator with spherical mirrors.

parallel to the resonator axis tends to remain inside it after multiple round trips then the resonator is stable. Conversely, in an unstable resonator a ray will tend to leave after a few round trips [63]. Using the ray-transfer matrix of a resonator and the self-consistency requirement for stability gives the condition for stability [64]:

06 g1g26 1, (1.1.1)

where mirror 1 has radius R1, mirror 2 has radius R2, L is the resonator length, and g1=

1 − L

R₁ and g2= 1 − L R₂.

This is illustrated in Fig. 1.2, where the resonator configurations yielding stable res-onators are shown in coloured areas, and unstable resres-onators in white areas.

1.1.2 Gaussian beams

A geometrical or ray transfer approach is useful to quantify the degree of stability of a resonator but does not predict the intensity distribution of a laser beam. Light in a res-onator is more accurately regarded as the electromagnetic field that is a solution of the one-dimensional wave equation. One important solution is the Gaussian intensity distribu-tion, which while not the only soludistribu-tion, is the most fundamental and the most commonly selected in commercial lasers. The Gaussian intensity distribution is illustrated in Fig. 1.3, and has the form:

I(r) = I0exp

−2r2

w2

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Figure 1.2: Plot of the stability function as a function of g1(x-axis) and g2(y-axis), with stability

in the coloured areas.

Figure 1.3: A Gaussian beam profile showing the beam radius w.

where I0is the peak intensity of the beam. w is the size of the laser beam and is defined

as the radius at which the beam intensity falls to 1/e2 (13.5 percent) of its peak value (see Fig. 1.3).

At some point along the axis of propagation (usually denoted z = 0) the beam has the smallest transverse extent, known as the waist, which is also the point at which the wave front is planar.

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laser Gaussian profile z=0 Plane wavefront _z=z Max. curvatureR z= Plane wavefront µ Gaussian profile 2w0

Figure 1.4: Propagation of a Gaussian laser beam.

curvature as they propagate (see Fig. 1.4) according to: w(z) = w0 " 1 + z zR 2#1₂ (1.1.3) and R(z) = z " 1 + zR z 2# , (1.1.4)

where z is the distance propagated from the plane with flat wave-front, w0is the 1/e2radius

of the beam waist, zR(= πw2₀/λ ) is the Rayleigh range, w(z) is the 1/e2beam radius at z,

and R(z) is the wave-front radius of curvature at z.

If the waist is at z = 0 (where R(z) is infinite), then as the beam propagates R(z) passes through a minimum at some z = zR, and increases again toward infinity with z.

Simultane-ously, the 1/e2intensity contours asymptotically approach a cone of angular radius :

θ = λ

π w0

. (1.1.5)

This is called the half-angle divergence of the Gaussian beam and is a measure of the diver-gence or transverse spread of the beam with distance.

Referring back to Fig. 1.1 and applying the steady-state condition that the radius of the phase front must be static at any arbitrary plane, and that it must match the radius of curvature of the mirrors, yields expressions for the Gaussian beam emerging from the resonator. The waist size is given by:

w4₀= λ π L(R1− L)(R2− L)(R1+ R2− L) (R1+ R2− 2L)2 (1.1.6) and is located at z1= L(R2− L) R1+ R2− 2L and z2= L(R1− L) R1+ R2− 2L (1.1.7)

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Gaussian propagation in the resonator gives the spot sizes on the mirrors: w4₁= λ R1 π 2 L(R2− L) (R1− L)(R1+ R2− L) and w4₂= λ R2 π 2 L(R1− L) (R2− L)(R1+ R2− L) (1.1.8)

A real resonator will always contain a limiting aperture of radius a, which might be the smallest dimension of the laser gain medium, a cavity mirror, or an intracavity iris. The Fresnel number is defined as:

NF=

a2

λ L (1.1.9)

where a is the limiting aperture radius, λ is the laser wavelength, and L is the resonator length.

The significance of the Fresnel number is that it defines the transverse extent available to the laser beam, which limits the number of higher-order modes which can be supported in a resonator. This is discussed in more detail in Section 1.1.4.

1.1.3 Beam quality

Many laser applications require a high-quality beam, in other words a beam with a defined cross-section that does not diverge too quickly. The Second Moment beam propagation ratio M2 is a common and widely-used parameter which summarizes the beam quality in one number [65]. According to this definition, the M2of a Gaussian beam is 1, and greater than one for all other beams.

1.1.4 Laser modes

The paraxial wave equation for the Fabry-Perot resonator can be solved by a number of complete and orthogonal sets of polynomials, which obey the orthogonality relationship [66]:

Z ∞ −∞

a(x)Ψm(x)Ψn(x)dx = δmn, (1.1.10)

where Ψn(x) is the set of polynomials, a(x) is a weighting function, and δnm is the

Kronecker delta.

The two most common basis sets for mathematically describing the intensity distribu-tion in a resonator are the Hermite and Laguerre polynomials, which describe the family

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of Hermite-Gauss (HG) and Laguerre-Gauss (LG) laser modes, respectively. In a resonator with no apertures there are infinitely many eigenmodes, and these are referred to as trans-verse electromagnetic (TEM) resonator modes. The HG modes tend to form in resonators with rectangular symmetry, while LG modes tend to form in resonators with circular sym-metry.

One important property of laser modes is that the intensity distribution is identical at any arbitrary plane along the optical axis inside (and outside) the resonator.

1.1.4.1 Hermite-Gaussian modes

One set of eigenmodes has the form of Hermite-Gaussian (HG) functions [67; 68] in rectan-gular coordinates and are denoted by TEM HGnm, where n is the order in the x−direction,

mis the order in the y−direction, and w is the beam radius of the associated TEM HG00or

Gaussian mode. These modes have an intensity distribution with the form [69]:

unm(x, y, z) = 1 w(ζ )Hm √ 2 x w(ζ ) Hn √ 2 y w(ζ ) exp ikz− ρ 2 w2₀(1 + iζ )− iΨm,n , (1.1.11) where k = 2π/λ is the wave number, w(z) is the beam size at longitudinal position z, w₀is the beam waist, zRis the Rayleigh range, ζ = z/zR, Ψm,n= (m + n + 1) arctan(ζ ), and

Hm, Hnare the Hermite polynomials are found using [70; 71]:

Hn(z) = (−1)nexp(z2)

dn

dznexp(−z

2_), _(1.1.12)

and the first few Hermite polynomials are given by:

H0(z) = 1

H1(z) = 2z

H2(z) = 4z2− 2

H3(z) = 8z3− 12z.

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In,m(x, y, z) = |un,m(x, y, z)|2 (1.1.13)

Figure 1.5: An array of plots of the TEM HG intensity distributions where the mode indices corre-spond to the horizontal, n and vertical, m index, respectively.

Fig. 1.5 shows transverse mode patterns for TEM HG modes of various orders.

w

₃

w

₅

Figure 1.6: The cross-section of a HG00 (blue), HG01 (purple) and HG02(yellow) mode, all with

the same w in 1.1.11.

Fig. 1.6 illustrates an important property of higher-order modes which is that the trans-verse extent of the modes increases with order. The spot sizes of higher-order modes in rectangular coordinates can be approximated by:

wn= w

√

n+ 1 and wm= w

√

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where w is the spot size of the corresponding TEM HG00 mode and m and n are the

orders of the x- and y-modes respectively.

For a Hermite-Gaussian mode, if wnmand w00are the waists of a high-order and

funda-mental beam respectively, then wnm(z) = Mw00(z) and

M_x2= 2n + 1 M_y2= 2m + 1

(1.1.15)

1.1.4.2 Laguerre-Gaussian modes

Another common set of eigenmodes can be expressed in cylindrical coordinates using La-guerre functions. The fields of the LaLa-guerre-Gaussian (LG) beams [72] are given by:

u`_p(r, φ , z) = s 2p! π (p + |`|!) 1 w(z)exp −r2 w2_(z)− ikr2 2R(z) L|`|p 2r2 w2_(z) × " √ 2r w(z) #|`| exp[i`φ ] exp −i(2p + |`| + 1) arctan z zR , (1.1.16)

where w(z) is the beam radius at longitudinal position z, p and ` are the radial and azimuthal indices, zR is the Rayleigh range, and L`p are the Laguerre polynomials, which

are the solutions of the differential equation:

xd 2_L` p dx2 + (` + 1 − x) dL`_p dx + pL ` p= 0, (1.1.17)

where the first few LG polynomials are given by: Ll₀(x) = 1 Ll₁(x) = ` + 1 − x Ll₂(x) =1 2(` + 1)(` + 2) − (` + 2)x+ = 1 2x 2_.

The intensity distribution of Laguerre-Gauss beams is given by:

I`_p(r, φ , z) =u`_p(r, φ , z)

2

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Figure 1.7: An array of plots of the LG intensity distributions where the mode indices correspond to the radial, p and azimuthal, ` index, respectively.

Fig. 1.7 shows transverse mode patterns for LG modes of various orders. Notice that the transverse extent of each mode increases with increasing order.

For a Laguerre-Gaussian mode defined by Eqn. 1.1.16 [73], the beam quality is found to be:

M2= 2p + ` + 1. (1.1.19)

1.1.5 Mode discrimination

In the absence of an intracavity aperture and any other obstructing elements, a Fabry-Perot resonator will have a fundamental mode with beam radius w1given by Eq. 1.1.8 on mirror

1 say, where w1≥ w2. The same resonator could, in theory, also produce the entire set of

Hermite-Gaussian (or Laguerre-Gaussian) modes as described by Eq. 1.1.11. As can be seen in Fig. 1.6 however, for a resonator with the same length L and mirror radii R1and R2,

that higher order modes are physically wider than lower order modes. Fig. 1.8 shows the transmission of several TEM orders as a function of the limiting aperture radius a on mirror 1. It can be seen that an aperture with a = 2w confers a 10% loss on the TEM22mode, but

insignificant losses to the TEM00 and TEM11modes. An aperture with a = 1.5w however

confers a 13% loss on the TEM11mode, and insignificant losses to the TEM00 modes. It

is clear that in general a smaller aperture will cut off a higher percentage of a higher order mode than a lower order mode.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 aperture radius a (cm) TEM00 TEM11 TEM22 TEM33 transmission w 1.5w 2w 2.5w 3w

Figure 1.8: Transmission values for several TEMnmmodes with w = 0.023 cm as a function of the

aperture size a. The vertical lines represent a = w, 1.5w, 2w, 2.5w, and 3w respectively.

the sum of the losses it experiences. Losses like the transmission through the output coupler or imperfect optics, for example, are the same for every mode, but an aperture will confer higher losses to higher order modes, thereby discriminating against these modes. The most common method of ensuring that a laser resonator produces a Gaussian beam is to insert an iris (an adjustable aperture) inside the resonator in front of one mirror, and to successively decrease the diameter of the iris. The laser will produce successively lower-order modes until the lowest-order mode, a Gaussian beam, is selected. Methods to produce higher-order modes and custom beams are described in Section 1.3.

1.1.6 Modal decomposition

It has been noted [65] that measuring the intensity profile of a laser beam is not a reliable method of identifying the modal composition of the beam. For example, a beam with an apparent Gaussian profile can be the sum of a number of non-Gaussian modal components, which will propagate very differently to a Gaussian beam. It is often necessary therefore to identify the components of a beam in terms of some complete, orthogonal polynomial set (see Eq. 1.1.10). The first method for the decomposition of transverse modes was devised in 1982, before phase-plate synthesis was possible [74]. Thereafter, methods devised for

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phase plates became easier and more convenient using a phase-only SLM. The optical inner product technique [75; 76] is one convenient method of performing modal decomposition. Using the basis set exp[i`φ ] as an example, a field u can be expressed in terms of these harmonics:

A beam which emerges from a Fabry-Perot resonator can be described by a linear com-bination of eigenmodes: U(r) = ∞

∑

n=1 anΨn(r), (1.1.20)

where Ψn(r) are eigenmodes, an are weighting coefficients, and U (r) is the resulting

output field. The coefficients ancan be determined using an inner product, given by:

an= hU, Ψni =

ZZ

R

U(r)Ψ∗_n(r) d2r, (1.1.21)

where the∗represents the complex conjugate. An arbitrary paraxial optical beam can be completely decomposed into the basis elements once the respective correlation coefficients have been determined. The weightings are optically determined by sampling the resultant field (u(x, y) = U (x, y)Ψ∗n(x, y)) in the Fourier plane where the corresponding Fourier

trans-formation is expressed as:

U1(kx, ky) =F {u(x,y)} =

ZZ

U(x, y)Ψ∗_n(x, y) exp[−i(kxx+ kyy)]dxdy. (1.1.22)

The weightings an can be found using an inner product, by measuring the on-axis

in-tensity of the field in the Fourier plane by setting the propagation vectors in 1.1.22 to zero (kx= ky= 0) to get: I(0, 0) = |U1(0, 0)|2= ZZ U(x, y)Ψ∗n(x, y)dxdy 2 = |an|2. (1.1.23)

1.2 Introduction to SLM technology

A spatial light modulator (SLM) refers to a device which spatially modulates coherent light [77]. They make use of liquid crystals, and are dynamically controlled by computers. There are two types of SLMs:

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• those which modulate the intensity of light, and are commonly used in computer-controlled projectors, and

• those which modulate phase (or phase and intensity simultaneously), and are used to modify the wave-front of laser beams in applications like laser tweezing, wave-front correction, and data processing.

Liquid crystals are used to modulate both intensity and phase. They are transparent rod-shaped molecules which align similarly to crystals but are free to slide across each other similarly to liquids. In the nematic phase, as used in SLMs, molecules are positioned randomly, but can be aligned by an applied electric field. Birefringence is another important property of liquid crystals, meaning that they have a different refractive index perpendicular to the optical axis (no) from parallel to it (ne), see Fig. 1.9. The property of birefringence is

denoted:

β = ne− no (1.2.1)

These molecules align themselves along an electric field, and therefore the optical axis can be rotated by modulating an electric field. In this way light passing through a liquid crystal layer can be slowed by between [no− 1]c and [ne− 1]c (where c is the speed if light),

resulting in a phase delay [78].

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There are two types of liquid crystals in use in SLMs, twisted nematic liquid crystals (TN-LCs) and parallel aligned liquid crystals (PA-LCs). The difference between these two types is show in Fig. 1.10.

Figure 1.10: Two examples of LC alignment schemes. (a) twisted nematic, (b) parallel aligned.

When a TN-LC layer is trapped between two sheets of glass, with no applied electric field, the molecules align to be parallel with the glass surfaces and with the optical axis of the crystals creating a twist or spiral through 90° from the top surface to the bottom surface. An applied electric field aligns the molecules to be perpendicular to the glass surfaces.

In a PA-LC layer however the molecules align parallel to the glass surfaces, and all point in the same direction. An applied electric field rotates all the molecules, keeping them parallel to each other, but perpendicular to the glass surfaces.

Both TN-LCs and PA-LCs are used in SLMs, with TN-LCs in older devices and PA-LCs in newer devices.

There are two basic types of SLMs [79]:

• an optically addressed (OA) SLM which uses incoherent light to map spatial modu-lation, and

• an electrically addressed (EA) SLM, which uses electrical signals to map spatial mod-ulation.

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1.2.1 Optically Addressed SLM (OASLM)

The basic OASLM (also known as a “light valve”) system is shown in Fig. 1.11.

Figure 1.11: Typical OASLM layout, showing that the phase pattern is written to the detector of the OASLM with “write light”, and is imparted onto the coherent “read light” by the modulator [1].

The OASLM works as follows: Incoherent light (or ‘write light’) is used as a signal, and a desired phase pattern is imaged onto the detector as a grey-scale image. The intensity of the write light is detected by a photo-detector and is converted to an electrical charge distribution. This charge distribution aligns the liquid crystal molecules in regions of higher intensity, which changes the phase of the coherent light (or ‘read light’) in these regions.

OASLMs are capable of forming large high-resolution holograms, and since they are not pixelated they avoid the two-dimensional grating effect found in EASLMs [80]. However, these devices suffer several disadvantages:

• It is difficult to keep the contrast and sensitivity across the device constant;

• The device is relatively insensitive to write light, and has a low contrast ratio of only 20:1;

• They tend to retain the written image;

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Table 1.1: Comparison of specifications of some commercially-available OASLMs.

Manufacturer Resolution Active area Reflectivity Switch frequency

(lp/mm) (mm) (%) (Hz)

Telecom Bretagne 100 35 × 45 > 85 2 − 3 × 10−3

Vavilov State Optical Institute 100 diam 30 − 45 unknown 100

University of Cambridge 825 16 × 21 unknown 0.1 − 10

For these reasons OASLMs are regarded as being experimental devices, and are there-fore very expensive. Some specifications of commercially-available OASLMs are shown in Table 1.1.

1.2.2 Electrically Addressed SLM (EASLM)

The basic EASLM system [81] is shown in Fig. 1.12. They combine liquid crystal technol-ogy with existing complementary metal oxide semiconductor (CMOS) technoltechnol-ogy, which has been used for many years for video display applications. The silicon back plane of the CMOS (Fig. 1.12(a)) consists of electronic circuitry under pixel arrays (b), and forms the substrate of the device. The pixels are reflective aluminium deposited on the silicon backplane. A cell consisting of LC material (c) trapped between two alignment layers (d) is placed on top of the reflective pixel layer. An indium tin oxide (ITO) layer (e) forms a transmissive electrode, and this layer is protected by a glass substrate (f). The circuitry in the CMOS allows a voltage to be applied to each pixel, which is used to alter the refractive index of the liquid crystal layer and thereby changing the phase delay of incident light (g) which is reflected and modulated (h) by the cell. The SLM device is attached to driver elec-tronics, and controlled by a computer as an additional display. The required phase is plotted to a bitmap image, the phase screen, the resolution of which matches the resolution of the SLM, and the required phase of each pixel mapped to 255 grey levels. This phase screen is displayed on the SLM device.

While SLMs can be used for a wide range of incident light wavelengths, it is important to note that the topmost glass substrate (Fig. 1.12(f)) is anti-reflection coated for a spe-cific wavelength band, which must be matched to the incident light. Also, because of the

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Table 1.2: Comparison of specifications of some commercially-available EASLMs. Manufacturer Resolution Active area Reflectivity Switch frequency

(lp/mm) (mm) (%) (Hz)

HoloEye 1920 × 1080 15.4 × 8.64 75 60

Hamamatsu 792 × 600 16 × 12 98 60

Boulder NLS 512 × 512 7.68 × 7.68 80 − 95 60

birefringent nature of LCs, SLMs only work as phase modulators for polarized light; the specific polarization direction is specified by the manufacturer.

Figure 1.12: Structure of an EASLM, showing a liquid crystal layer (c) sandwiched between two alignment layers (d) on an aluminium pixel array (b) which is mounted on a CMOS chip (a). A voltage between the CMOS chip and in indium tin oxide layer (e) controls the birefringence of the liquid crystals in each pixel, which changes the phase of the incident light (g) to obtain modulated, reflected light (h) [1].

The most common experimental setup in order to use an SLM for beam-shaping is shown in Fig. 1.13. A polarized laser beam must be used, which is orientated to match the SLM polarization direction. A beam expanding telescope (BET) is often necessary to match the beam size to the size of the SLM. A collimated beam with flat wave-front is reflected off the SLM displaying the required phase screen, thereby acquiring the required phase modulation. Since an SLM is a repetitive structure, several orders of diffraction will be present, separated by a characteristic angle. If, as is often the case, the far-field pattern of the modulated beam is required, then an iris is used to block off unwanted orders and only

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allow the selected order through to be captured on a camera placed in the Fourier plane of a lens.

Figure 1.13: Common experimental setup of an SLM to generate custom beams. Laser light is polarized and then expanded onto the SLM. One diffraction order is selected using an iris at the focal plane of a lens and recorded on a camera (CAM).

1.3 Custom modes and laser beam shaping

Soon after the invention of the laser in 1960 [82], distinctive intensity patterns in the beams or modes were modelled using round-trip loss considerations [83]. By 1962 scientists had inserted a circular aperture into a resonator to select for the lowest-order mode, the Gaussian beam [3; 84], and by 1972 the first-order T EM01was selected for as the output beam [85].

Hermite-Gaussian beams were reasonably easy to obtain, but Laguerre-Gauss beams proved to be more difficult. They were first preferentially selected in 1990 using a pump-shaping technique [86].

Several techniques have been devised for selecting higher-order modes. The first and simplest was to insert fine metal wires near one of the end mirrors coinciding with node lines which give high loss to all but the the desired mode, and was demonstrated for both HG modes [87] and LG modes [88]. Wires work by introducing scattering losses, but heat up and become inefficient. An alternative is to use non-absorbing phase elements, which

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introduce losses by interference and diffraction. A phase plate with a π phase shift line acts as a loss line inside a resonator, but with higher mode discrimination [6]. A π phase shift is used so that the there is a 2π phase shift after one round trip through the resonator, resulting in no net phase shift. This concept can be expanded to multiple lines or rings to select low-order HG or LG modes [89; 90]. The discovery that LG modes have wave-fronts with nπ spiral phase shifts [91] can be exploited to create LG0

nmodes by inserting a 2π spiral phase

plate into the resonator [92]. Different spiral beams have also been created by inserting a Dove prism into a ring resonator in order to rotate the internal beam [93].

Pump-shaping provides regions of high and low gain inside a resonator and can thus function in a similar way to inserted regions of loss. The pump intensity profile can be modulated by, for example diffraction from an aperture [94], or by a phase-plate [95].

It is generally true that beam-shaping methods can be applied to any wavelength of laser light, although some are more suited to longer wavelengths due to challenges in man-ufacturing processes. For example, custom-shaped or aspherical mirrors have been used to shape the output beam [96], but this method is limited to lasers with longer wavelengths like CO2 lasers because of relatively large feature size. Deformable mirrors are active

as-pherical mirrors, able to respond to changes in wave-front using a wave-front sensor in a closed loop. These have been used first to improve beam quality [97], for the formation of a super-Gaussian output beam [98], for aberration correction [99], and to dynamically switch between various low- and high-order output modes [100].

Modulating the phase inside a resonator allows the formation of new and custom in-tensity distributions. One common method is using diffractive optical elements (DOEs), which are etched using an electron beam using a multi-stage mask process. Intracavity DOEs have been used for intracavity beam-shaping to generate, for example, a square flat-top beam [7]. Holographic elements have been recorded in photo-polymer sheets, with an SLM to control the phase of a signal beam, and used as intracavity elements to generate super-Gaussian output beams from an Nd:YAG laser [101]. Phase modulation has also been provided by an intracavity optically-activated SLM, and used to produce circular and square super-Gaussian beams [102]. However this design has the disadvantage of requiring a complicated optical imaging system to address the OA SLM.

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never before been used as a phase modulating element inside a laser resonator. The design consideration necessary to achieve this are discussed.

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1.4 Outline

This thesis deals with the the use of SLMs in the field of novel laser beams. It is structured as follows.

An EA-SLM was used for the first time as an intracavity optical element in order to achieve real-time intracavity beam shaping. The details of how this was achieved is ex-plained in Chapter 2.

An SLM was used to generate Laguerre-Gauss beams, which have self-healing proper-ties. The radial flow of energy which results in self-healing is explained, with experimental verification, in Chapter 3.

An SLM was used to generate completely new beams, belonging to a class of Bessel-like beams, which also have self-healing properties resulting from a longitudinal energy flow. This is explained, with experimental verification, in Chapter 4.

Finally, this thesis is concluded in Chapter 5 with a summary of our contributions to the field of beam shaping using an SLM, and a discussion of future work within this field.

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Chapter 2

SLM for intracavity beam shaping

In this chapter we outline the steps necessary to create a laser with an intra-cavity spatial light modulator (SLM) for transverse mode control. We employ a commercial SLM as the back reflector in an otherwise conventional diode-pumped solid state laser. We show that the geometry of the liquid crystal (LC) arrangement strongly influences the operation regime of the laser, from nominally amplitude-only mode control for twisted nematic LCs to nominally phase-only mode control for parallel-aligned LCs. We demonstrate both oper-ating regimes experimentally and discuss the potential advantages of and improvements to this new technology.

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2.1 Introduction

Good beam quality associated with lower-order modes is a fundamental requirement for industrial applications like cutting and welding that require a tightly focused beam. Appli-cations such as paint stripping, penetration laser drilling and thin-film welding, however, require a flat-top beam profile, while high-volume parallel processes require a single beam to be split into an array of beams. The required beam shape for a particular application may be created by a range of techniques [8].

For example, a simple amplitude filter may be used to produce a Gaussian beam, but at the expense of power. A more efficient method of manipulating the intensity distribution of a given beam is using phase plates, but these are static, custom components, and their performance deteriorates with any variation in size of the initial beam [103]. Deformable mirrors were originally developed to correct for atmospheric disturbance in telescopes, but have proved useful for beam shaping applications, and have been used for producing circular and rectangular flat-top intensity profiles. They have the drawback however that the number of mirror elements is limited, and so the feature size of the beams produced by deformable mirrors is therefore limited [99; 104]. A more common approach today is to use liquid crystal displays in the form of spatial light modulators (SLMs) to dynamically mimic both amplitude and phase transformations. These devices are easily programmed by simply displaying the required phase, represented by a bitmap image, on the high-resolution SLM screen [54].

For the most part the aforementioned techniques are used to modify an existing beam outside a resonator, but it is possible to reduce the number of optical elements and increase the efficiency of a system by putting the modulating device inside the resonator. Intra-cavity amplitude filters, phase plates and deformable mirrors have all been used to modify the output beam [105; 106]. An intracavity optically addressed SLM has also been used to manipulate the beam intensity profile [101], but required a complex intracavity imaging sys-tem to create a phase screen. (Intracavity mode selection and custom modes are discussed in more detail in Section 1.3). More recently we have demonstrated the on-demand creation of modes with an intracavity electrically addressed SLM [55; 57; 58; 60]. The unique ad-vantages of using an intracavity electrically-addressed SLM are the ability to create a very wide range of free-space beams, and the ability to do so dynamically.

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In this chapter we outline the necessary steps to construct a laser incorporating an in-tracavity electrically addressed SLM for transverse mode selection. We outline the design considerations, advantages and disadvantages of this approach, and provide a detailed per-formance evaluation of the SLM and laser. This work can be a useful reference for others wishing to build such devices.

2.1.1 Liquid crystal considerations

In a typical SLM cell a thin layer of liquid crystal (LC) material is sandwiched between transparent electrodes. The LC materials used in SLMs are birefringent, with two refractive indices that depend on the direction of the molecular axis. The LCs are also electro-active, and align according to the applied electric field. The most important configurations are twisted nematic (TN), and parallel aligned (PA) cells. In a twisted cell, the orientation of the molecules differs by typically 90° between the top and the bottom of the LC cell and rotates helically between (see 1.10 (left)). In PA cells, the alignment layers are parallel, so the LC molecules are oriented in the same plane (see 1.10 (right)).

The birefringence of the liquid crystals causes a change in the polarization of monochro-matic, polarized incident light, leading to a modulation of phase and/or amplitude. For TN cells the situation is complex, and a 2 × 2 Jones matrix is used to model the change in po-larization of light passing through a TN LC cell, the eigenvectors of which correspond to elliptically polarized waves that propagate through the system without a change in the po-larization state, and are subject only to phase modulation [22] [107] [108]. A small degree of amplitude modulation is introduced by a polarizer behind the SLM, leading to the mode of operation which is often referred to as phase-mostly operation [109]. Fig. 2.1 is derived from reference [109], and shows that varying the incident polarizer angle can at best reduce the amplitude modulation of a field passing through a TN-LC cell, but never eliminate it completely.

In PA cells, light polarized linearly parallel to the extraordinary axis of the LC material is retarded as a function of the birefringence. Therefore, these cells are true phase-only modulators of linearly polarized light, with no amplitude modulation.

It is impossible to specify the chemical composition of the liquid crystal used in com-mercial SLMs, since this is always comcom-mercially confidential. It is usually a mixture,

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en-Figure 2.1: Theoretically calculated curves show the transmission through a TN-LC layer for twist angle 90 °, for incident polarizer angles 0°; 5°; 10° (solid line), 15° and 20°. The solid line shows the configuration with the most constant transmission [109].

gineered by a chemical company (Merck, for example), and suitable for the application. There will however be several key features that are common for this application, and the mixture will be optimised for them [110]. Firstly, both TN and PA LCs have a nematic phase, so specifying this is an obvious starting point. Secondly, a birefringence of around 0.2 is fairly typical; 5CB is just one well-known material with this property. Thirdly, LC mixtures need to have little temperature dependence at and around room temperature, so are chosen to undergo a transition to an isotropic liquid at temperatures greater than 60°C, and often as high as 100°C. It is for this reason that LC cannot be used at high powers. Fourthly, the response time (typically 10 ms) depends on the viscosity, elastic constants, device thickness and applied voltage. Also, a compromise must be found between a small cell gap which gives a faster response, and a thicker gap which is slower but gives a wider phase modulation range.

2.2 SLM characterisation and design considerations

In standard operation, each pixel on an SLM is addressed by a pixel on a grey-scale bitmap. For each pixel a grey scale level between 0 and 255 corresponds to a phase change of between 0° and 360° being imparted on the reflected beam by the corresponding pixel on the SLM.

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Table 2.1: Comparison of typical specifications of the older type of SLM using TN-LCs and the newer type using PA-LCs.

SLM type Resolution (pixels) Area (mm) LC Reflectivity Damage Threshold (W/cm2)

TN-LC 1920 × 1080 15.36 × 8.64 TN ∼ 60% 2

PA-LC 792 × 600 16 × 12 PA > 90% 15

As will be seen in the discussion that follows, the most significant differences in per-formance of SLMs used as an intracavity component were a result of the type of liquid crystal used in the SLM. We consider here the two most common liquid crystal geometries: twisted-nematic liquid crystals (TN-LC), and parallel-aligned liquid crystals (PA-LC). Ta-ble (2.1) shows a comparison of typical specifications.

For most LCD applications a high resolution is regarded as being desirable. For SLMs used inside a resonator, however, the lower resolution of the PA-LC SLM presented no limitations.

Both types of SLM require linear vertically polarized light to perform optimally as phase screens, and behave as plane mirrors for light polarized perpendicular to this axis. It is therefore necessary to ensure vertical polarization in all experiments, and in the design of the SLM resonator. In addition, the possibility that either SLM would depolarize the beam was also considered. Experiments confirm however that there is no depolarization of the incident light on a single reflection off the SLMs for any phase.

Since the intracavity power is typically one order of magnitude higher than the extra-cavity power, one of the primary considerations is to prevent damage to the SLM. Depend-ing on the expected intracavity power density it could be necessary to expand the beam in order to decrease the power density. Any clipping of the beam by the edges of the SLM active area will result in distortion of the desired mode. Using the second moment defini-tion of beam radius, as a starting point the expected beam radius should be designed to be between1_/₄_and1_/₆_{of the shorter dimension of the SLM active area. For example, the SLM}

with active area 15.36 × 8.64 mm would be illuminated by a spot radius no larger than 2.16 mm. An intracavity telescope is typically used to achieve this.

The zeroth-order reflectivity of the TN-LC SLM and PA-LC SLM were specified to be 60% and > 90% respectively, with no specification given as to the variation in reflectivity

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with phase. The reflectivity of each SLM as a function of phase R(θ ) was measured ex-perimentally by reflecting a 1.064 µm Nd:YAG laser beam of constant power off the SLM and recording the reflected power. The grey level or phase shift of a uniform screen was increased in steps from 0 to 255 grey shades, or from 0° to 360°. The mean reflectivity of the TN-LC SLM over 360° was measured to be 51% for vertical (correct) polarization and 64% for horizontal (incorrect) polarization. The mean reflectivity of the PA-LC SLM over 360° was measured to be 91% for vertical (correct) polarization and 93% for horizontal (in-correct) polarization. One result of this is that a laser with this intracavity component will tend to produce radiation with polarization which is incorrect for the SLM, and that another polarization-selecting component needs to be included to ensure the correct polarization. Typically a Brewster window is used inside a cavity to select one polarization direction.

In conventional use, the beam reflected off an SLM is at a small angle from the inci-dent beam, and a phase grating superimposed on the desired phase serves to separate the diffracted light away from the undiffracted light. Since it replaces a mirror, an SLM in-side a resonator must be aligned perpendicular to the optical axis, with the reflected beam returning along the path of the incident beam, with the diffracted and undiffracted beams coaxial. Since a laser preferentially amplifies the mode with lowest loss, it tends to amplify the mode which is selected for by the SLM screen, and suppress any other modes, includ-ing that containinclud-ing the undiffracted beam. In a similar way, as it is a pixelated device an SLM has the property of any periodic structure in that a small fraction of incident light is diffracted into higher orders. Fortunately these higher orders are diffracted away from the optical axis and lost, and only the lowest order containing the selected mode is amplified.

The average reflectivity of an SLM acts as a loss in the cavity, which can be compen-sated for with higher gain, typically by increasing the pump power. Far more important for our application is the variation in reflectivity with phase. The percentage variation in reflec-tivity was measured at 9.5% and 0.75% for the TN-LC SLM and PA-LC SLM respectively, as in Figure 2.2.

The explanation for this variation in reflectivity can be found in reference [109], which explains that in TN-LC modulators with no field applied, the liquid crystal molecules are aligned in a 90° spiral between the front and back of the layer. When an electric field is applied across the layer, the liquid crystal molecules become tilted and cause an ellipticity

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0.4 0.5 0.6 0.7 0.8 0.9 1 0 60 120 180 240 300 360 phase (degrees) TN-LC PA-LC reflectivity

Figure 2.2: Measured reflectivities of TN-LC SLM and PA-LC SLM as a function of phase, for vertical polarization.

in the polarization state. In TN-LC SLMs, the transmitted intensity is determined by the LC properties of twist angle and birefringence, as well as the orientation angles of internal polarizers which are used to cut out the non-linear component. The result is an unavoidable amplitude modulation that can be reduced but not eliminated altogether (compare Fig. 2.1 to Fig. 2.2 (TN-LC curve)), and therefore TN-LC SLMs are referred to in the literature as “phase-only” or phase-mostly SLMs [109; 111; 112].

Remember however that SLMs are designed to be used as single reflectors, and in typ-ical applications this residual amplitude modulation is negligible. Inside a laser resonator, however, it will be shown to have a determining effect on the laser performance. In PA-LC devices, the liquid crystal molecules are aligned in parallel, not in spirals, when no electric field is applied. When an electric field is applied the molecules are tilted in the direction of the substrates. When the polarization direction of the incident light is parallel to the axis of the liquid crystal molecules only the refractive index along the optic axis is changed, the light is not depolarized, and in theory phase-only modulation can be achieved [30; 113; 114]. In practice a small residual amplitude modulation can be measured, caused by reflection changes at the optical boundary and diffraction due to index changes of the LC material [115].

If R1(θ ) is the reflectivity of the SLM, and R2is the reflectivity of the output coupler,

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given by

I(θ , n) = (R1(θ )R2)n. (2.2.1)

This simple amplification model reveals that this small variation in SLM amplitude modulation, when amplified through many round trips in a resonator, is sufficient to cause a higher lasing threshold at some phase values than at others. The model uses the amplitude modulation data from Figure 2.2, and is used to generate the graphs in Figure 2.3 which predict relative intensity as a function of phase after 0, 5, 10 and 20 round trips for the TN-LC SLM and the PA-LC SLM resonators respectively.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 60 120 180 240 300 360 phase (degrees) 20 round trips 0 5 10 15 relativeintensity( a.u .) (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 60 120 180 240 300 360 phase (degrees) 0 5 10 15 20 round trips relativeintensity( a.u .) (b)

Figure 2.3: Plots of normalized relative intensity as a function of phase after n = 0, 5, 10 and 20 round-trip reflections of a beam with stable mode off an SLM in a resonator, using measured intensity modulation data for (a) a TN-LC SLM and (b) a PA-LC SLM.

Figure 2.3 (a) suggests that when a TN-LC SLM is used as an intracavity element, the variation in amplitude modulation which accompanies the desired phase modulation

Novel implementation of a phase-only spatial light modulator for laser beam shaping

Liesl Burger

Declaration

Abstract

Uittreksel

Acknowledgements

Dedications

Contents

List of Figures

List of Tables

List of Publications

Peer-reviewed Journal Papers:

Patents:

International Conference Papers:

National Conference Papers:

Nomenclature

List of symbols

Acronyms/Abbreviations

Chapter 1

Introduction

1.1

Basic laser theory

R

R

L

2ù

z

z

2ù

2ù

w

w

w

∑

1.2

Introduction to SLM technology

1.3

Custom modes and laser beam shaping

1.4

Outline

Chapter 2

SLM for intracavity beam shaping

2.1

Introduction

2.2

SLM characterisation and design considerations

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